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GRAVITY“You can’t blame gravity for falling in love.” — Albert Einstein
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GRAVITY
LITTLE PLANET
▸ If an asteroid was very small but supermassive, could you really live on it like the Little Prince?
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GRAVITY
WHAT IS GRAVITY?▸ Gravity is geometry
▸ Distortions in spacetime due to the presence of mass
▸ Spacetime is the four dimensional fabric of the Universe
▸ It’s a mathematical model that combines the three spatial dimensions (up/down, left/right, forward/backward) and time into a single, interwoven continuum
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GRAVITY
WHAT IS GRAVITY?
▸ Means by which masses communicate with each other
▸ Masses attract each other
▸ Equal and opposite
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GRAVITY
Four Fundamental Forces of Fysics
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GRAVITY
SIR ISAAC NEWTON AND THE 1ST LAW OF MOTION
▸ ca 1680s
▸ Galileo introduced us to inertia in 16th Century
▸ Newton’s 1st Law of Motion:
▸ Object at rest tends to stay at rest unless acted on by an outside force
▸ What about falling objects?
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GRAVITY
ISAAC NEWTON
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GRAVITY
FALLING MOON
▸ Why doesn’t the Moon crash into Earth?
▸ The Moon has tangential velocity
▸ What if the force of gravity disappeared?
▸ The Moon would fly away in a straight, tangential path
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GRAVITY
ORBITS
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FLYING IS JUST FALLING AND MISSING THE GROUND
Douglas Adams
GRAVITY
ORBITS
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GRAVITY
ORBITS
▸ Weight on a string:tension::orbital motion:__________________
▸ Same force that acts on all falling objects
the force of gravity
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GRAVITY
UNIVERSAL GRAVITY
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GRAVITY
FORMALIZING GRAVITY▸ The force of gravity
between two objects depends on 3 quantities
▸ How big is the first object: m1
▸ How big is the second object: m2
▸ How far apart are they: r
m1
m2
Fg
Fg
r
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GRAVITY
INVERSE SQUARE LAW
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GRAVITY
INVERSE SQUARE LAW
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GRAVITY
INVERSE SQUARE LAW
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GRAVITY
INVERSE SQUARE LAW
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GRAVITY
FORCE OF GRAVITY
▸ Fg = Gm1m2
▸ G = 6.67×10-11 Nm2/kg2
▸ Gravitational constant
▸ Always the same, no matter where you are in the Universe
r2
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GRAVITY
FALLING MOON TEST
▸ Mass doesn’t matter. Only distance
▸ Compare fall of an apple to fall of the Moon
▸ Apple falls 4.9m in 1s
▸ The Moon is 60x further from Earth
▸ Fg is diluted by 1/r2
▸ Moon falls 1.4mm — QED
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GRAVITY
EXAMPLE 1
▸ What’s the force of gravity between two students?
▸ m1 = ______________
▸ m2 = ______________
▸ r = ______________
▸ Ans: Fg = ____________
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GRAVITY
THE MASS OF EARTH
▸ F = ma
▸ mg = GmM/R2
▸ M = gR2/G
▸ R = 6.37x106m
▸ M = (9.81m/s2)(6.37x106m)2/(6.67x10-11Nm2/kg2)
▸ M = 5.97x1024kg
= GmM/R2
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GRAVITY
GRAVITATIONAL POTENTIAL ENERGY
▸ PEg = -Gm1m2r
▸ Why negative?
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GRAVITY
EXAMPLE 2▸ How fast would you need to throw
a ball of mass m for it to escape Earth’s gravity?
▸ E = KE + PE
▸ E = ½mv2 - GMm/R = 0
▸ Ans. vesc = √ 2GM/R
▸ Escape speed
▸ Independent of the mass of the object in question!
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Mass RadiusSun 1.99×1030 kg 6.96×108 m
Mercury 3.20×1023 kg 2.44×106 mVenus 4.88×1024 kg 6.05×106 mEarth 5.97×1024 kg 6.37×106 mMars 6.42×1023 kg 3.40×106 m
Jupiter 1.90×1027 kg 7.15×107 mSaturn 5.68×1026 kg 6.03×107 mUranus 8.68×1025 kg 2.56×107 m
Neptune 1.02×1026 kg 2.48×107 m
GRAVITY
OUR SOLAR SYSTEM
Earth 5.97×1024 kg 6.37×106 m
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GRAVITY
EXAMPLE 3
▸ What’s escape speed from the surface of Earth?
▸ Ans. vesc = 11.2 km/s
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GRAVITY
EXAMPLE 4
▸ What’s the orbital velocity at a distance r away from the center of Earth?
▸ Fg = Fc
▸ Ans. vorb = √ GM/R
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GRAVITY DID YOU GET ALL THAT?
▸ Why don’t the planets crash into the Sun?
▸ What if vt = 0m/s?
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GRAVITY
PULLING UP ON THE EARTH
▸ What is the force of gravity between the Earth and a 70.0kg student?
▸ m = 70.0 kg M = 5.97x1024 kg
▸ R = 6.37x106 m G = 6.67x10-11 Nm2/kg2
▸ Fg = GmM/R2
▸ Fg = (6.67x10-11Nm2/kg2)(70.0kg)(5.97x1024kg)
▸ Fg = 687 N
(6.37x106m)2
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GRAVITY
PULLING UP ON THE EARTH
▸ Fg = 687 N
▸ What is the acceleration of the Earth due to this student?
▸ F = Ma
▸ 687N = (5.97x1024 kg) a
▸ a = 1.15x10-22 m/s2
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GRAVITY
HERE’S A QUESTION
▸ An apple on a tree feels 1N of force due to gravity
▸ If the tree were twice as tall, the force of gravity on the apple would then be:
A. twice as strong
B. half as strong
C. a fourth as strong
D. none of the above
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GRAVITY
HERE’S A QUESTION
▸ An apple on a tree feels 1N of force due to gravity
▸ If the tree were twice as tall, the force of gravity on the apple would then be:
A. twice as strong
B. half as strong
C. a fourth as strong
D. none of the above
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GRAVITY
ALTITUDE
▸ What is the force of gravity at 2.53x105m above the surface of the Earth on a 50.0kg object?
▸ M = 5.97x1024kg
▸ R = 6.37x106m
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GRAVITY
ALTITUDE
▸ Fg = GmM/r2
▸ r = R + h
▸ r = 6.37x106m + 2.53x105m
▸ r = 6.623x106m
▸ Fg = (6.67x10-11Nm2/kg2)(50.0kg)(5.97x1024kg)
▸ Fg = 454N
(6.623x106m)2
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GRAVITY
ALTITUDE
▸ Fg = 454N
▸ What is the normal weight on the surface of Earth?
▸ W = mg
▸ W = (50.0kg)(9.81m/s2)
▸ W = 491N
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GRAVITY
“WEIGHTLESSNESS”
▸ Astronauts in orbit are said to be “weightless”
▸ Gravity is still acting on them!
▸ Astronauts are in perpetual free fall!
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GRAVITY
DOUBLE CHECK
▸ Since the Moon is pulled to the Earth by gravitational attraction, why doesn’t it crash into the Earth?
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GRAVITY
DOUBLE CHECK
▸ If there is an attractive force between all objects, why do we not feel ourselves gravitating toward massive buildings in our vicinity?
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GRAVITY
DOUBLE CHECK
▸ Which requires more fuel — a rocket going from from Earth to the Moon or a rocket coming from the Moon to Earth? Why?
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GRAVITY
SATELLITE MOTION▸ Shape of orbit?
▸ Ellipse
▸ What’s conservation of angular momentum tell us about the planet’s orbit?
▸ The closer it gets, the faster it orbits
▸ L = mr12ω1 = mr22ω2
Fastest
Slowest
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GRAVITY
EARTH’S ORBIT
▸ At its furthest, Earth is 1.52×1011 m from the Sun. At that distance, our planet orbits at 1.93×10-7 rad/s. At its nearest, Earth is 1.47×1011 m from the Sun. What’s its orbital velocity at that distance (in rad/s)?
▸ Ans. ω = 2.06×10-7 rad/s
▸ What’s that orbital velocity in m/s?
▸ Ans. v = 30,300 m/s
▸ That’s 88 times the speed of sound! (or 73 times a speeding bullet!)
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GRAVITY
LITTLE PLANET
▸ If an asteroid was very small but supermassive, could you really live on it like the Little Prince?
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GRAVITY
LITTLE PLANET▸ If the astroid has a radius of
1.75 m, what mass would it need to in order to have Earthlike gravity at the surface?
▸ M = 4.5×1011 kg (just under 500 million tons)
▸ That’s roughly equal to the combined mass of every human on Earth
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GRAVITY
LITTLE PLANET▸ What would be the difference between the force of gravity at your
head vs. your feet?
▸ Compare the force of gravity on 60 kg at the surface vs. 1.6 m (~5’ 3”) above the surface?
▸ Fg (h = 0) = 588 N
▸ Fg (h = 1.6 m) = 160 N
▸ Called tidal forces — where there’s a substantial difference in the force of gravity between different ends of a body
▸ It would feel like lying on a merry-go-round with your head near the center
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GRAVITY
LITTLE PLANET▸ What would be the escape speed
on the surface?
▸ vesc = 5.9 m/s
▸ That’s slower than a sprint (but still pretty fast)
▸ If you can’t dunk a basketball, you wouldn’t be able to escape the asteroid by jumping
▸ However, you might be able to escape by running horizontally and jumping off a ramp
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GRAVITY
LITTLE PLANET
▸ Say you center of mass is 1.4 m above your feet. How fast would you need to run to start orbiting the asteroid?
▸ vorb = 3.1 m/s
▸ That’s typical jogging speed
▸ This would be a weird orbit
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HISTORY OF GRAVITATIONAL THEORY
“If I have seen further than others, it is by standing on the shoulders of giants” — Isaac Newton
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GRAVITY
SCIENTIFIC LAW VS. THEORY
https://www.youtube.com/watch?v=GyN2RhbhiEU
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GRAVITY
ERATOSTHENES CA. 276 BC- 194 BC
▸ Greek mathematician, geographer, poet, astronomer, and music theorist
▸ First to measure the circumference of the Earth
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GRAVITY
THE CIRCUMFERENCE OF THE EARTH▸ Syene lies on the Tropic of
Cancer
▸ Noon on the Summer Solstice, the Sun is directly overhead
▸ At the that same time, measured the Sun’s angle of elevation in Alexandria
▸ 7º 12’ (or 1/50th of a circle)
▸ Calculated circumference: 46,620km (err. 16.3%)
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GRAVITY
REASONS FOR ERROR
▸ Alexandria and Syene do not lie on the same meridian
▸ In accurate distance between Alexandria and Syene
▸ Earth is not a perfect sphere
▸ Accepted circumference of Earth: 40,075km
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GRAVITY
PTOLEMY CA. 100 - 170 AD
▸ Greco-Egyptian mathematician, astronomer, geographer, astrologer, and poet
▸ Developed his Planetary Hypotheses which provided the conventional model of the Universe for the next millennium and a half
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GRAVITY
THE PTOLEMAIC MODEL OF THE UNIVERSE
▸ Consisted of concentric shells, called celestial spheres, with the Earth at the center
▸ Geocentric Theory
▸ The Sun and other planets orbit the Earth
▸ The stars are fixed to the outmost shell
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GRAVITY
ORBITS UNDER GEOCENTRIC THEORY
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GRAVITY
NICOLAUS COPERNICUS CA. 1473 - 1543
▸ Prussian mathematician and astronomer
▸ Father of the Heliocentric Theory
▸ Earth goes around the Sun, not vise versa
▸ Often cited as the beginning of the Scientific Revolution
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GRAVITY
TYCHO BRAHE
CA. 1546 - 1601
▸ Danish astronomer, astrologer, and alchemist
▸ Published De Nova Stella in 1573
▸ Discovered that our Universe is dynamic and ever changing
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GRAVITY
A DYNAMIC UNIVERSE
▸ In 1572, observed a bright new star in the constellation Cassiopeia
▸ Lacked a daily parallax against the background of stars, suggesting it was much further away
▸ Parallax - difference in apparent position of an object viewed along two different lines of sight
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GRAVITY
PARALLAX
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GRAVITY
A DYNAMIC UNIVERSE
▸ In 1572, observed a bright new star in the constellation Cassiopeia
▸ Lacked a daily parallax against the background of stars, suggesting it was much further away
▸ Parallax - difference in apparent position of an object viewed along two different lines of sight
‣ Refuted the Aristotelian belief in an unchanging celestial realm
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GRAVITY
JOHANNES KEPLER CA. 1571 - 1630▸ German mathematician,
astronomer, and astrologer
▸ Assistant to Tycho Brahe
▸ Laws of Planetary Motion
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GRAVITY
KEPLER’S THREE LAWS OF PLANETARY MOTION
1) The orbit of a planet is an ellipse with the Sun at one of the two focus points
2) A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time
3) The square of the orbital period of a planet is proportional to the cube of its average distance from the Sun
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GRAVITY
KEPLER’S 1ST LAW
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GRAVITY
KEPLER’S 2ND LAW
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GRAVITY
KEPLER’S 2ND LAW
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GRAVITY
KEPLER’S 3RD LAW
‣ T2 = 4π2r3
‣ r = the average distance from the sunGMS
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GRAVITY
EXAMPLE 8
▸ Determine the mass of the Sun given the Earth is an average distance of 1.5×1011 m from the Sun.
▸ Ans. MS = 2.0×1030 kg
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GRAVITY
LAWS OF PLANETARY MOTION, REDUCED
1) Planets orbit in an ellipse
2) The change in area swept out by a planet per time is always constant
3) T2 = 4π2r3
GMS
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GRAVITY
KEPLER’S KONTRIBUTIONS
▸ Improved heliocentric theory
▸ Planets move in ellipses, not circles
▸ The Sun is at a focal point, not the center
▸ Neither linear speed nor angular speed are constant. Area speed in constant
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GRAVITY
GALILEO GALILEI CA. 1564 - 1642
▸ Italian physicist, mathematician, engineer, astronomer, and philosopher
▸ In the absence of air resistance, all objects accelerate toward the Earth at the same rate, regardless of mass
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GRAVITY
ISAAC NEWTON CA. 1642 - 1726▸ English physicist and
mathematician
▸ Published Principia in 1687
▸ Laws of Motion
▸ Foundation of classical mechanics
▸ Law of Universal Gravitation
▸ Derivation of Kepler’s Laws of Planetary Motion
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GRAVITY
NEWTON’S THEORY OF GRAVITY▸ Explained the orbit of the planets
▸ Explained the motion of falling objects
▸ In 1821, Alexis Bouvard observed disturbances in the orbit of Uranus
▸ Newton’s theory of gravity predicted these disturbances were due to the presence of an unknown gravitational source and eventually led to the discovery of Neptune
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GRAVITY
NINTH PLANET?
https://www.caltech.edu/news/caltech-researchers-find-evidence-real-ninth-planet-49523
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GRAVITY
NEWTON’S THEORY OF GRAVITY▸ However, Newton’s theory
could not accurately account for Mercury’s eccentric orbit
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GRAVITY
ALBERT EINSTEIN CA. 1879 - 1955▸ German theoretical physicist
and philosopher of science
▸ Gravity as curvature of spacetime
▸ Einstein’s theory of general relativity explained the precession of Mercury’s orbit is due to the severely curved spacetime near the Sun
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1.The speed of light is the same in all inertial reference frames
Albert Einstein
GRAVITY
EINSTEIN’S SPECIAL THEORY OF RELATIVITY
c = 299,792,458 m/s (186,282 mi/s)
c = 3.00×108 m/s
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2.The laws of physics are the same in all inertial reference frames
Albert Einstein
GRAVITY
EINSTEIN’S SPECIAL THEORY OF RELATIVITY
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GRAVITY
IMPLICATIONS‣ Nothing can go faster than light
‣ If the Sun magically vanished, how long would it take Earth to know?
‣ Length and time are malleable — they change depending on your reference frame
‣ The closer you move to the speed of light, the more time dilates and length contracts
‣ https://www.youtube.com/watch?v=CYv5GsXEf1o
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GRAVITY
EINSTEIN’S GENERAL THEORY RELATIVITY▸ Gravity is the universal
attraction between all mass, and, since E = mc2, all forms of energy
▸ Gravity is the curvature of spacetime — the four dimensional fabric of the universe
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GRAVITY
BENDING SPACETIME
https://www.youtube.com/watch?v=MTY1Kje0yLg
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GRAVITY
SCHWARZSCHILD METRIC
▸ ds2 = - 1 - 2GM (c dt)2 + 1 - 2GM -1 dr2 + r2 (dθ2 + sin2θ dφ2)
▸ Describes the geometry of spacetime around a star
▸ Super important in GPS navigation
( ( ))c2r c2r
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GRAVITY
EDWIN HUBBLE CA. 1889 - 1953▸ American astronomer and
cosmologist
▸ Universe goes beyond the Milky Way galaxy
▸ Expanding Universe
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GRAVITY
ANDROMEDA GALAXY
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GRAVITY
ANDROMEDA GALAXY
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GRAVITY EXPANDING UNIVERSE
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GRAVITY
EXPANDING UNIVERSE; RED SHIFT
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